Intrepid
http://trilinos.sandia.gov/packages/docs/r10.12/packages/intrepid/test/Discretization/Basis/HGRAD_HEX_C1_FEM/test_02.cpp
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00038 //                    Denis Ridzal  (dridzal@sandia.gov), or
00039 //                    Kara Peterson (kjpeter@sandia.gov)
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00042 // @HEADER
00043 
00049 #include "Intrepid_FieldContainer.hpp"
00050 #include "Intrepid_HGRAD_HEX_C1_FEM.hpp"
00051 #include "Intrepid_DefaultCubatureFactory.hpp"
00052 #include "Intrepid_RealSpaceTools.hpp"
00053 #include "Intrepid_ArrayTools.hpp"
00054 #include "Intrepid_FunctionSpaceTools.hpp"
00055 #include "Intrepid_CellTools.hpp"
00056 #include "Teuchos_oblackholestream.hpp"
00057 #include "Teuchos_RCP.hpp"
00058 #include "Teuchos_GlobalMPISession.hpp"
00059 #include "Teuchos_SerialDenseMatrix.hpp"
00060 #include "Teuchos_SerialDenseVector.hpp"
00061 #include "Teuchos_LAPACK.hpp"
00062 
00063 using namespace std;
00064 using namespace Intrepid;
00065 
00066 void rhsFunc(FieldContainer<double> &, const FieldContainer<double> &, int, int, int);
00067 void neumann(FieldContainer<double>       & ,
00068              const FieldContainer<double> & ,
00069              const FieldContainer<double> & ,
00070              const shards::CellTopology   & ,
00071              int, int, int, int);
00072 void u_exact(FieldContainer<double> &, const FieldContainer<double> &, int, int, int);
00073 
00075 void rhsFunc(FieldContainer<double> & result,
00076              const FieldContainer<double> & points,
00077              int xd,
00078              int yd,
00079              int zd) {
00080 
00081   int x = 0, y = 1, z = 2;
00082 
00083   // second x-derivatives of u
00084   if (xd > 1) {
00085     for (int cell=0; cell<result.dimension(0); cell++) {
00086       for (int pt=0; pt<result.dimension(1); pt++) {
00087         result(cell,pt) = - xd*(xd-1)*std::pow(points(cell,pt,x), xd-2) *
00088                             std::pow(points(cell,pt,y), yd) * std::pow(points(cell,pt,z), zd);
00089       }
00090     }
00091   }
00092 
00093   // second y-derivatives of u
00094   if (yd > 1) {
00095     for (int cell=0; cell<result.dimension(0); cell++) {
00096       for (int pt=0; pt<result.dimension(1); pt++) {
00097         result(cell,pt) -=  yd*(yd-1)*std::pow(points(cell,pt,y), yd-2) *
00098                             std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,z), zd);
00099       }
00100     }
00101   }
00102 
00103   // second z-derivatives of u
00104   if (zd > 1) {
00105     for (int cell=0; cell<result.dimension(0); cell++) {
00106       for (int pt=0; pt<result.dimension(1); pt++) {
00107         result(cell,pt) -=  zd*(zd-1)*std::pow(points(cell,pt,z), zd-2) *
00108                             std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd);
00109       }
00110     }
00111   }
00112 
00113   // add u
00114   for (int cell=0; cell<result.dimension(0); cell++) {
00115     for (int pt=0; pt<result.dimension(1); pt++) {
00116       result(cell,pt) +=  std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd) * std::pow(points(cell,pt,z), zd);
00117     }
00118   }
00119 
00120 }
00121 
00122 
00124 void neumann(FieldContainer<double>       & result,
00125              const FieldContainer<double> & points,
00126              const FieldContainer<double> & jacs,
00127              const shards::CellTopology   & parentCell,
00128              int sideOrdinal, int xd, int yd, int zd) {
00129 
00130   int x = 0, y = 1, z = 2;
00131 
00132   int numCells  = result.dimension(0);
00133   int numPoints = result.dimension(1);
00134 
00135   FieldContainer<double> grad_u(numCells, numPoints, 3);
00136   FieldContainer<double> side_normals(numCells, numPoints, 3);
00137   FieldContainer<double> normal_lengths(numCells, numPoints);
00138 
00139   // first x-derivatives of u
00140   if (xd > 0) {
00141     for (int cell=0; cell<numCells; cell++) {
00142       for (int pt=0; pt<numPoints; pt++) {
00143         grad_u(cell,pt,x) = xd*std::pow(points(cell,pt,x), xd-1) *
00144                             std::pow(points(cell,pt,y), yd) * std::pow(points(cell,pt,z), zd);
00145       }
00146     }
00147   }
00148 
00149   // first y-derivatives of u
00150   if (yd > 0) {
00151     for (int cell=0; cell<numCells; cell++) {
00152       for (int pt=0; pt<numPoints; pt++) {
00153         grad_u(cell,pt,y) = yd*std::pow(points(cell,pt,y), yd-1) *
00154                             std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,z), zd);
00155       }
00156     }
00157   }
00158 
00159   // first z-derivatives of u
00160   if (zd > 0) {
00161     for (int cell=0; cell<numCells; cell++) {
00162       for (int pt=0; pt<numPoints; pt++) {
00163         grad_u(cell,pt,z) = zd*std::pow(points(cell,pt,z), zd-1) *
00164                             std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd);
00165       }
00166     }
00167   }
00168   
00169   CellTools<double>::getPhysicalSideNormals(side_normals, jacs, sideOrdinal, parentCell);
00170 
00171   // scale normals
00172   RealSpaceTools<double>::vectorNorm(normal_lengths, side_normals, NORM_TWO);
00173   FunctionSpaceTools::scalarMultiplyDataData<double>(side_normals, normal_lengths, side_normals, true); 
00174 
00175   FunctionSpaceTools::dotMultiplyDataData<double>(result, grad_u, side_normals);
00176 
00177 }
00178 
00180 void u_exact(FieldContainer<double> & result, const FieldContainer<double> & points, int xd, int yd, int zd) {
00181   int x = 0, y = 1, z = 2;
00182   for (int cell=0; cell<result.dimension(0); cell++) {
00183     for (int pt=0; pt<result.dimension(1); pt++) {
00184       result(cell,pt) = std::pow(points(pt,x), xd)*std::pow(points(pt,y), yd)*std::pow(points(pt,z), zd);
00185     }
00186   }
00187 }
00188 
00189 
00190 
00191 
00192 int main(int argc, char *argv[]) {
00193 
00194   Teuchos::GlobalMPISession mpiSession(&argc, &argv);
00195 
00196   // This little trick lets us print to std::cout only if
00197   // a (dummy) command-line argument is provided.
00198   int iprint     = argc - 1;
00199   Teuchos::RCP<std::ostream> outStream;
00200   Teuchos::oblackholestream bhs; // outputs nothing
00201   if (iprint > 0)
00202     outStream = Teuchos::rcp(&std::cout, false);
00203   else
00204     outStream = Teuchos::rcp(&bhs, false);
00205 
00206   // Save the format state of the original std::cout.
00207   Teuchos::oblackholestream oldFormatState;
00208   oldFormatState.copyfmt(std::cout);
00209 
00210   *outStream \
00211     << "===============================================================================\n" \
00212     << "|                                                                             |\n" \
00213     << "|                    Unit Test (Basis_HGRAD_HEX_C1_FEM)                       |\n" \
00214     << "|                                                                             |\n" \
00215     << "|     1) Patch test involving mass and stiffness matrices,                    |\n" \
00216     << "|        for the Neumann problem on a physical parallelepiped                 |\n" \
00217     << "|        AND a reference hex Omega with boundary Gamma.                       |\n" \
00218     << "|                                                                             |\n" \
00219     << "|        - div (grad u) + u = f  in Omega,  (grad u) . n = g  on Gamma        |\n" \
00220     << "|                                                                             |\n" \
00221     << "|        For a generic parallelepiped, the basis recovers a complete          |\n" \
00222     << "|        polynomial space of order 1. On a (scaled and/or translated)         |\n" \
00223     << "|        reference hex, the basis recovers a complete tensor product          |\n" \
00224     << "|        space of order 1 (i.e. incl. xy, xz, yz, xyz term).                  |\n" \
00225     << "|                                                                             |\n" \
00226     << "|  Questions? Contact  Pavel Bochev  (pbboche@sandia.gov),                    |\n" \
00227     << "|                      Denis Ridzal  (dridzal@sandia.gov),                    |\n" \
00228     << "|                      Kara Peterson (kjpeter@sandia.gov).                    |\n" \
00229     << "|                                                                             |\n" \
00230     << "|  Intrepid's website: http://trilinos.sandia.gov/packages/intrepid           |\n" \
00231     << "|  Trilinos website:   http://trilinos.sandia.gov                             |\n" \
00232     << "|                                                                             |\n" \
00233     << "===============================================================================\n"\
00234     << "| TEST 1: Patch test                                                          |\n"\
00235     << "===============================================================================\n";
00236 
00237   
00238   int errorFlag = 0;
00239 
00240   outStream -> precision(16);
00241 
00242 
00243   try {
00244 
00245     int max_order = 1;                                                                    // max total order of polynomial solution
00246     DefaultCubatureFactory<double>  cubFactory;                                           // create factory
00247     shards::CellTopology cell(shards::getCellTopologyData< shards::Hexahedron<> >());     // create parent cell topology
00248     shards::CellTopology side(shards::getCellTopologyData< shards::Quadrilateral<> >());  // create relevant subcell (side) topology
00249     int cellDim = cell.getDimension();
00250     int sideDim = side.getDimension();
00251     unsigned numSides = 6;
00252 
00253     // Define array containing points at which the solution is evaluated, on the reference tet.
00254     int numIntervals = 10;
00255     int numInterpPoints = (numIntervals + 1)*(numIntervals + 1)*(numIntervals + 1);
00256     FieldContainer<double> interp_points_ref(numInterpPoints, 3);
00257     int counter = 0;
00258     for (int k=0; k<=numIntervals; k++) {
00259       for (int j=0; j<=numIntervals; j++) {
00260         for (int i=0; i<=numIntervals; i++) {
00261           interp_points_ref(counter,0) = i*(1.0/numIntervals)-1.0;
00262           interp_points_ref(counter,1) = j*(1.0/numIntervals)-1.0;
00263           interp_points_ref(counter,2) = k*(1.0/numIntervals)-1.0;
00264           counter++;
00265         }
00266       }
00267     }
00268 
00269     /* Parent cell definition. */
00270     FieldContainer<double> cell_nodes[2];
00271     cell_nodes[0].resize(1, 8, cellDim);
00272     cell_nodes[1].resize(1, 8, cellDim);
00273 
00274     // Generic parallelepiped.
00275     cell_nodes[0](0, 0, 0) = -5.0;
00276     cell_nodes[0](0, 0, 1) = -1.0;
00277     cell_nodes[0](0, 0, 2) = 0.0;
00278     cell_nodes[0](0, 1, 0) = 4.0;
00279     cell_nodes[0](0, 1, 1) = 1.0;
00280     cell_nodes[0](0, 1, 2) = 1.0;
00281     cell_nodes[0](0, 2, 0) = 8.0;
00282     cell_nodes[0](0, 2, 1) = 3.0;
00283     cell_nodes[0](0, 2, 2) = 1.0;
00284     cell_nodes[0](0, 3, 0) = -1.0;
00285     cell_nodes[0](0, 3, 1) = 1.0;
00286     cell_nodes[0](0, 3, 2) = 0.0;
00287     cell_nodes[0](0, 4, 0) = 5.0;
00288     cell_nodes[0](0, 4, 1) = 9.0;
00289     cell_nodes[0](0, 4, 2) = 1.0;
00290     cell_nodes[0](0, 5, 0) = 14.0;
00291     cell_nodes[0](0, 5, 1) = 11.0;
00292     cell_nodes[0](0, 5, 2) = 2.0;
00293     cell_nodes[0](0, 6, 0) = 18.0;
00294     cell_nodes[0](0, 6, 1) = 13.0;
00295     cell_nodes[0](0, 6, 2) = 2.0;
00296     cell_nodes[0](0, 7, 0) = 9.0;
00297     cell_nodes[0](0, 7, 1) = 11.0;
00298     cell_nodes[0](0, 7, 2) = 1.0;
00299     // Reference hex.
00300     cell_nodes[1](0, 0, 0) = -1.0;
00301     cell_nodes[1](0, 0, 1) = -1.0;
00302     cell_nodes[1](0, 0, 2) = -1.0;
00303     cell_nodes[1](0, 1, 0) = 1.0;
00304     cell_nodes[1](0, 1, 1) = -1.0;
00305     cell_nodes[1](0, 1, 2) = -1.0;
00306     cell_nodes[1](0, 2, 0) = 1.0;
00307     cell_nodes[1](0, 2, 1) = 1.0;
00308     cell_nodes[1](0, 2, 2) = -1.0;
00309     cell_nodes[1](0, 3, 0) = -1.0;
00310     cell_nodes[1](0, 3, 1) = 1.0;
00311     cell_nodes[1](0, 3, 2) = -1.0;
00312     cell_nodes[1](0, 4, 0) = -1.0;
00313     cell_nodes[1](0, 4, 1) = -1.0;
00314     cell_nodes[1](0, 4, 2) = 1.0;
00315     cell_nodes[1](0, 5, 0) = 1.0;
00316     cell_nodes[1](0, 5, 1) = -1.0;
00317     cell_nodes[1](0, 5, 2) = 1.0;
00318     cell_nodes[1](0, 6, 0) = 1.0;
00319     cell_nodes[1](0, 6, 1) = 1.0;
00320     cell_nodes[1](0, 6, 2) = 1.0;
00321     cell_nodes[1](0, 7, 0) = -1.0;
00322     cell_nodes[1](0, 7, 1) = 1.0;
00323     cell_nodes[1](0, 7, 2) = 1.0;
00324 
00325     std::stringstream mystream[2];
00326     mystream[0].str("\n>> Now testing basis on a generic parallelepiped ...\n");
00327     mystream[1].str("\n>> Now testing basis on the reference hex ...\n");
00328 
00329 
00330     for (int pcell = 0; pcell < 2; pcell++) {
00331       *outStream << mystream[pcell].str();
00332       FieldContainer<double> interp_points(1, numInterpPoints, cellDim);
00333       CellTools<double>::mapToPhysicalFrame(interp_points, interp_points_ref, cell_nodes[pcell], cell);
00334       interp_points.resize(numInterpPoints, cellDim);
00335 
00336       for (int x_order=0; x_order <= max_order; x_order++) {
00337         int max_y_order = max_order;
00338         if (pcell == 0) {
00339           max_y_order -= x_order;
00340         }
00341         for (int y_order=0; y_order <= max_y_order; y_order++) {
00342           int max_z_order = max_order;
00343           if (pcell == 0) {
00344             max_z_order -= x_order;
00345             max_z_order -= y_order;
00346           }
00347           for (int z_order=0; z_order <= max_z_order; z_order++) {
00348 
00349             // evaluate exact solution
00350             FieldContainer<double> exact_solution(1, numInterpPoints);
00351             u_exact(exact_solution, interp_points, x_order, y_order, z_order);
00352 
00353             int basis_order = 1;
00354 
00355             // set test tolerance;
00356             double zero = basis_order*basis_order*basis_order*100*INTREPID_TOL;
00357 
00358             //create basis
00359             Teuchos::RCP<Basis<double,FieldContainer<double> > > basis =
00360               Teuchos::rcp(new Basis_HGRAD_HEX_C1_FEM<double,FieldContainer<double> >() );
00361             int numFields = basis->getCardinality();
00362 
00363             // create cubatures
00364             Teuchos::RCP<Cubature<double> > cellCub = cubFactory.create(cell, 2*basis_order);
00365             Teuchos::RCP<Cubature<double> > sideCub = cubFactory.create(side, 2*basis_order);
00366             int numCubPointsCell = cellCub->getNumPoints();
00367             int numCubPointsSide = sideCub->getNumPoints();
00368 
00369             /* Computational arrays. */
00370             /* Section 1: Related to parent cell integration. */
00371             FieldContainer<double> cub_points_cell(numCubPointsCell, cellDim);
00372             FieldContainer<double> cub_points_cell_physical(1, numCubPointsCell, cellDim);
00373             FieldContainer<double> cub_weights_cell(numCubPointsCell);
00374             FieldContainer<double> jacobian_cell(1, numCubPointsCell, cellDim, cellDim);
00375             FieldContainer<double> jacobian_inv_cell(1, numCubPointsCell, cellDim, cellDim);
00376             FieldContainer<double> jacobian_det_cell(1, numCubPointsCell);
00377             FieldContainer<double> weighted_measure_cell(1, numCubPointsCell);
00378 
00379             FieldContainer<double> value_of_basis_at_cub_points_cell(numFields, numCubPointsCell);
00380             FieldContainer<double> transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
00381             FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
00382             FieldContainer<double> grad_of_basis_at_cub_points_cell(numFields, numCubPointsCell, cellDim);
00383             FieldContainer<double> transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
00384             FieldContainer<double> weighted_transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
00385             FieldContainer<double> fe_matrix(1, numFields, numFields);
00386 
00387             FieldContainer<double> rhs_at_cub_points_cell_physical(1, numCubPointsCell);
00388             FieldContainer<double> rhs_and_soln_vector(1, numFields);
00389 
00390             /* Section 2: Related to subcell (side) integration. */
00391             FieldContainer<double> cub_points_side(numCubPointsSide, sideDim);
00392             FieldContainer<double> cub_weights_side(numCubPointsSide);
00393             FieldContainer<double> cub_points_side_refcell(numCubPointsSide, cellDim);
00394             FieldContainer<double> cub_points_side_physical(1, numCubPointsSide, cellDim);
00395             FieldContainer<double> jacobian_side_refcell(1, numCubPointsSide, cellDim, cellDim);
00396             FieldContainer<double> jacobian_det_side_refcell(1, numCubPointsSide);
00397             FieldContainer<double> weighted_measure_side_refcell(1, numCubPointsSide);
00398 
00399             FieldContainer<double> value_of_basis_at_cub_points_side_refcell(numFields, numCubPointsSide);
00400             FieldContainer<double> transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
00401             FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
00402             FieldContainer<double> neumann_data_at_cub_points_side_physical(1, numCubPointsSide);
00403             FieldContainer<double> neumann_fields_per_side(1, numFields);
00404 
00405             /* Section 3: Related to global interpolant. */
00406             FieldContainer<double> value_of_basis_at_interp_points_ref(numFields, numInterpPoints);
00407             FieldContainer<double> transformed_value_of_basis_at_interp_points_ref(1, numFields, numInterpPoints);
00408             FieldContainer<double> interpolant(1, numInterpPoints);
00409 
00410             FieldContainer<int> ipiv(numFields);
00411 
00412 
00413 
00414             /******************* START COMPUTATION ***********************/
00415 
00416             // get cubature points and weights
00417             cellCub->getCubature(cub_points_cell, cub_weights_cell);
00418 
00419             // compute geometric cell information
00420             CellTools<double>::setJacobian(jacobian_cell, cub_points_cell, cell_nodes[pcell], cell);
00421             CellTools<double>::setJacobianInv(jacobian_inv_cell, jacobian_cell);
00422             CellTools<double>::setJacobianDet(jacobian_det_cell, jacobian_cell);
00423 
00424             // compute weighted measure
00425             FunctionSpaceTools::computeCellMeasure<double>(weighted_measure_cell, jacobian_det_cell, cub_weights_cell);
00426 
00428             // Computing mass matrices:
00429             // tabulate values of basis functions at (reference) cubature points
00430             basis->getValues(value_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_VALUE);
00431 
00432             // transform values of basis functions 
00433             FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_cell,
00434                                                             value_of_basis_at_cub_points_cell);
00435 
00436             // multiply with weighted measure
00437             FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_cell,
00438                                                         weighted_measure_cell,
00439                                                         transformed_value_of_basis_at_cub_points_cell);
00440 
00441             // compute mass matrices
00442             FunctionSpaceTools::integrate<double>(fe_matrix,
00443                                                   transformed_value_of_basis_at_cub_points_cell,
00444                                                   weighted_transformed_value_of_basis_at_cub_points_cell,
00445                                                   COMP_BLAS);
00447 
00449             // Computing stiffness matrices:
00450             // tabulate gradients of basis functions at (reference) cubature points
00451             basis->getValues(grad_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_GRAD);
00452 
00453             // transform gradients of basis functions 
00454             FunctionSpaceTools::HGRADtransformGRAD<double>(transformed_grad_of_basis_at_cub_points_cell,
00455                                                            jacobian_inv_cell,
00456                                                            grad_of_basis_at_cub_points_cell);
00457 
00458             // multiply with weighted measure
00459             FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_grad_of_basis_at_cub_points_cell,
00460                                                         weighted_measure_cell,
00461                                                         transformed_grad_of_basis_at_cub_points_cell);
00462 
00463             // compute stiffness matrices and sum into fe_matrix
00464             FunctionSpaceTools::integrate<double>(fe_matrix,
00465                                                   transformed_grad_of_basis_at_cub_points_cell,
00466                                                   weighted_transformed_grad_of_basis_at_cub_points_cell,
00467                                                   COMP_BLAS,
00468                                                   true);
00470 
00472             // Computing RHS contributions:
00473             // map cell (reference) cubature points to physical space
00474             CellTools<double>::mapToPhysicalFrame(cub_points_cell_physical, cub_points_cell, cell_nodes[pcell], cell);
00475 
00476             // evaluate rhs function
00477             rhsFunc(rhs_at_cub_points_cell_physical, cub_points_cell_physical, x_order, y_order, z_order);
00478 
00479             // compute rhs
00480             FunctionSpaceTools::integrate<double>(rhs_and_soln_vector,
00481                                                   rhs_at_cub_points_cell_physical,
00482                                                   weighted_transformed_value_of_basis_at_cub_points_cell,
00483                                                   COMP_BLAS);
00484 
00485             // compute neumann b.c. contributions and adjust rhs
00486             sideCub->getCubature(cub_points_side, cub_weights_side);
00487             for (unsigned i=0; i<numSides; i++) {
00488               // compute geometric cell information
00489               CellTools<double>::mapToReferenceSubcell(cub_points_side_refcell, cub_points_side, sideDim, (int)i, cell);
00490               CellTools<double>::setJacobian(jacobian_side_refcell, cub_points_side_refcell, cell_nodes[pcell], cell);
00491               CellTools<double>::setJacobianDet(jacobian_det_side_refcell, jacobian_side_refcell);
00492 
00493               // compute weighted face measure
00494               FunctionSpaceTools::computeFaceMeasure<double>(weighted_measure_side_refcell,
00495                                                              jacobian_side_refcell,
00496                                                              cub_weights_side,
00497                                                              i,
00498                                                              cell);
00499 
00500               // tabulate values of basis functions at side cubature points, in the reference parent cell domain
00501               basis->getValues(value_of_basis_at_cub_points_side_refcell, cub_points_side_refcell, OPERATOR_VALUE);
00502               // transform 
00503               FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_side_refcell,
00504                                                               value_of_basis_at_cub_points_side_refcell);
00505 
00506               // multiply with weighted measure
00507               FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_side_refcell,
00508                                                           weighted_measure_side_refcell,
00509                                                           transformed_value_of_basis_at_cub_points_side_refcell);
00510 
00511               // compute Neumann data
00512               // map side cubature points in reference parent cell domain to physical space
00513               CellTools<double>::mapToPhysicalFrame(cub_points_side_physical, cub_points_side_refcell, cell_nodes[pcell], cell);
00514               // now compute data
00515               neumann(neumann_data_at_cub_points_side_physical, cub_points_side_physical, jacobian_side_refcell,
00516                       cell, (int)i, x_order, y_order, z_order);
00517 
00518               FunctionSpaceTools::integrate<double>(neumann_fields_per_side,
00519                                                     neumann_data_at_cub_points_side_physical,
00520                                                     weighted_transformed_value_of_basis_at_cub_points_side_refcell,
00521                                                     COMP_BLAS);
00522 
00523               // adjust RHS
00524               RealSpaceTools<double>::add(rhs_and_soln_vector, neumann_fields_per_side);;
00525             }
00527 
00529             // Solution of linear system:
00530             int info = 0;
00531             Teuchos::LAPACK<int, double> solver;
00532             solver.GESV(numFields, 1, &fe_matrix[0], numFields, &ipiv(0), &rhs_and_soln_vector[0], numFields, &info);
00534 
00536             // Building interpolant:
00537             // evaluate basis at interpolation points
00538             basis->getValues(value_of_basis_at_interp_points_ref, interp_points_ref, OPERATOR_VALUE);
00539             // transform values of basis functions 
00540             FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_interp_points_ref,
00541                                                             value_of_basis_at_interp_points_ref);
00542             FunctionSpaceTools::evaluate<double>(interpolant, rhs_and_soln_vector, transformed_value_of_basis_at_interp_points_ref);
00544 
00545             /******************* END COMPUTATION ***********************/
00546         
00547             RealSpaceTools<double>::subtract(interpolant, exact_solution);
00548 
00549             *outStream << "\nRelative norm-2 error between exact solution polynomial of order ("
00550                        << x_order << ", " << y_order << ", " << z_order
00551                        << ") and finite element interpolant of order " << basis_order << ": "
00552                        << RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
00553                           RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) << "\n";
00554 
00555             if (RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
00556                 RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) > zero) {
00557               *outStream << "\n\nPatch test failed for solution polynomial order ("
00558                          << x_order << ", " << y_order << ", " << z_order << ") and basis order " << basis_order << "\n\n";
00559               errorFlag++;
00560             }
00561           } // end for z_order
00562         } // end for y_order
00563       } // end for x_order
00564     } // end for pcell
00565 
00566   }
00567   // Catch unexpected errors
00568   catch (std::logic_error err) {
00569     *outStream << err.what() << "\n\n";
00570     errorFlag = -1000;
00571   };
00572 
00573   if (errorFlag != 0)
00574     std::cout << "End Result: TEST FAILED\n";
00575   else
00576     std::cout << "End Result: TEST PASSED\n";
00577 
00578   // reset format state of std::cout
00579   std::cout.copyfmt(oldFormatState);
00580 
00581   return errorFlag;
00582 }